Axis of rotation, plane of reflection

In summary, for a 3x3 orthogonal matrix with determinant=-1, the axis of rotation is the same as the plane of reflection. This is because the matrix can be written as a product of two unitary matrices, one of which has only real entries, resulting in a rotation and reflection around the same axis. However, it is necessary to prove that this matrix is proportional to a real matrix.
  • #1
JamesGoh
143
0
For a 3x3 orthogonal matrix with determinant= -1 (which means rotation followed by simple reflection), is the axis of rotation the same as the plane of reflection ?

My reasoning is follows (see attachment)

Say you have two vectors with the same angle size (which i call A), same x-values, but one of the z-values (height in this case), is the negative of the other (n.b. y-value is zero in both vectors )

Vector 1 rotates around the x-axis by 2A to get to the same spot as vector 2. Because the angle size is the same and because the z-component of vector 2 is the negative of the z-component of vector 1, we get a reflection ?

Since the "reflection" happens about the x-axis, this is why the plane of reflection is the same as the axis of rotation in the case of a 3x3 orthogonal matrix having determinant 1 ?
 

Attachments

  • examplepicture.jpg
    examplepicture.jpg
    4.1 KB · Views: 495
Physics news on Phys.org
  • #2
if [itex]A\in \textrm{O}(3,\mathbb{R})[/itex] and [itex]\det(A)=-1[/itex], there exists a [itex]U\in \textrm{SU}(3)[/itex] such that

[tex]
UAU^{\dagger} = \left(\begin{array}{ccc}
-1 & 0 & 0 \\
0 & e^{i\theta} & 0 \\
0 & 0 & e^{-i\theta} \\
\end{array}\right)
[/tex]

with some [itex]\theta\in\mathbb{R}[/itex].

Then there exists a [itex]V\in \textrm{SU}(2)[/itex] such that

[tex]
V\left(\begin{array}{cc}
e^{i\theta} & 0 \\
0 & e^{-i\theta} \\
\end{array}\right)V^{\dagger}
= \left(\begin{array}{cc}
\cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta) \\
\end{array}\right)
[/tex]

So if you define

[tex]
W = \left(\begin{array}{cc}
1 & 0 \\
0 & V \\
\end{array}\right)U
[/tex]

then [itex]WAW^{\dagger}[/itex] will be of such form that reflection and rotation are clearly carried out with respect to the same axis. Only problem is that [itex]W[/itex] doesn't necessarily have only real entries. How to prove that [itex]W[/itex] is necessarily proportional to a real matrix?
 

Related to Axis of rotation, plane of reflection

What is the definition of "axis of rotation"?

The axis of rotation is an imaginary line around which an object rotates or spins.

What is the significance of the "axis of rotation" in physics?

The axis of rotation is significant in physics because it determines the direction and magnitude of an object's angular velocity and angular momentum.

How is the "axis of rotation" related to angular motion?

The axis of rotation is directly related to angular motion, as it is the line that an object rotates around and experiences rotation in relation to.

What is meant by "plane of reflection"?

The plane of reflection is an imaginary surface that divides an object into two equal halves, with one half being the mirror image of the other half.

How does the "plane of reflection" affect the symmetry of an object?

The plane of reflection is a line of symmetry for an object, meaning that the object can be reflected across this plane and appear identical on both sides.

Similar threads

Replies
31
Views
2K
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
4
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
1K
  • Linear and Abstract Algebra
Replies
4
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
80
  • Linear and Abstract Algebra
Replies
2
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
2K

Back
Top